ASSIGNMENT 4 – PROPERTIES OF AXIOMATIC SYSTEMS
The 3 important properties of an axiomatic system are consistency, independence, and
completeness.
Consistency: An axiomatic system is said to be consistent if no two statements (could be
two axioms, and axiom and a theorem, or two theorems) contradict each other.
Example of an inconsistent axiomatic system:
Axioms:
1. There are exactly 3 points
2. There are exactly 2 points
Independence: An individual axiom in an axiomatic system is called independent if it
cannot be proved from the other axioms.
Example of an axiomatic system with an axiom that is not independent:
Axioms:
1.
2.
3.
4.
5.
There are exactly 3 points.
Two distinct points belong to one and only one line.
Not all points belong to the same line
Two separate lines have at least one point in common.
A line has exactly two points.
Axiom 5 is not independent. Prove!
Completeness: An axiomatic system is called complete if it is impossible to add a new
consistent and independent axiom to the system. The new axiom can use only defined
and undefined terms of the original system.
Example:
Undefined terms – points, lines
Axioms:
1. There are exactly 3 points
2. Two distinct points belong to one and only one line.
Now consider the additional statement “ Not all of the points belong to the same line”.
This statement is consistent. Also, the statement is not provable true or false within the
system consisting of axioms 1. and 2 and hence the system is not complete.
Model
Three-Point Geometry: Consider example 2 in Assignment 3. Replace ‘class’ and
‘students’ by ‘line’ and ‘point’ respectively. Interpret ‘line’ to be one of the segments
shown below, and ‘point’ to be any one of the three endpoints of the segments shown in
the figure.
A
B
C
Then, the axioms in Example 2, Assignment 3, would read as:
a)
b)
c)
d)
There are exactly 3 points
Two distinct points belong to one and only one line
Not all points belong to the same line
Two separate lines have at least one point in common
Exercise: State the theorems proved in Example 2, Assignment 3, in the new
interpretation.
This new interpretation of the original axiomatic system is called a model of the system.
Four-point geometry: The undefined terms are the same as in 3-point geometry. The
axioms are:
a) There are exactly 4 points
b) Two distinct points belong to one and only one line
c) Each line has exactly two points belonging to it.
Exercises:
1. Show there are exactly 6 lines in 4-point geometry
2. Show that each point belongs to exactly 3 lines,
3. Show that the 4-point geometry is relatively consistent to Euclidean geometry
4. If we define the vertex of a tetrahedron to be a point and a line to be an edge, will the
tetrahedron be a model for 4-point geometry? Why or why not?
Additional exercises to do: Page 33, #s 1.5.8, 1.5.9, 1.5.10